Bitcoin ABC  0.22.12
P2P Digital Currency
group_impl.h
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1 /**********************************************************************
2  * Copyright (c) 2013, 2014 Pieter Wuille *
3  * Distributed under the MIT software license, see the accompanying *
4  * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
5  **********************************************************************/
6 
7 #ifndef SECP256K1_GROUP_IMPL_H
8 #define SECP256K1_GROUP_IMPL_H
9 
10 #include "num.h"
11 #include "field.h"
12 #include "group.h"
13 
14 /* These exhaustive group test orders and generators are chosen such that:
15  * - The field size is equal to that of secp256k1, so field code is the same.
16  * - The curve equation is of the form y^2=x^3+B for some constant B.
17  * - The subgroup has a generator 2*P, where P.x=1.
18  * - The subgroup has size less than 1000 to permit exhaustive testing.
19  * - The subgroup admits an endomorphism of the form lambda*(x,y) == (beta*x,y).
20  *
21  * These parameters are generated using sage/gen_exhaustive_groups.sage.
22  */
23 #if defined(EXHAUSTIVE_TEST_ORDER)
24 # if EXHAUSTIVE_TEST_ORDER == 13
26  0xc3459c3d, 0x35326167, 0xcd86cce8, 0x07a2417f,
27  0x5b8bd567, 0xde8538ee, 0x0d507b0c, 0xd128f5bb,
28  0x8e467fec, 0xcd30000a, 0x6cc1184e, 0x25d382c2,
29  0xa2f4494e, 0x2fbe9abc, 0x8b64abac, 0xd005fb24
30 );
32  0x3d3486b2, 0x159a9ca5, 0xc75638be, 0xb23a69bc,
33  0x946a45ab, 0x24801247, 0xb4ed2b8e, 0x26b6a417
34 );
35 # elif EXHAUSTIVE_TEST_ORDER == 199
36 static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
37  0x226e653f, 0xc8df7744, 0x9bacbf12, 0x7d1dcbf9,
38  0x87f05b2a, 0xe7edbd28, 0x1f564575, 0xc48dcf18,
39  0xa13872c2, 0xe933bb17, 0x5d9ffd5b, 0xb5b6e10c,
40  0x57fe3c00, 0xbaaaa15a, 0xe003ec3e, 0x9c269bae
41 );
42 static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(
43  0x2cca28fa, 0xfc614b80, 0x2a3db42b, 0x00ba00b1,
44  0xbea8d943, 0xdace9ab2, 0x9536daea, 0x0074defb
45 );
46 # else
47 # error No known generator for the specified exhaustive test group order.
48 # endif
49 #else
50 
53 static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
54  0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL,
55  0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL,
56  0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL,
57  0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL
58 );
59 
60 static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 7);
61 #endif
62 
63 static void secp256k1_ge_set_gej_zinv(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zi) {
64  secp256k1_fe zi2;
65  secp256k1_fe zi3;
66  secp256k1_fe_sqr(&zi2, zi);
67  secp256k1_fe_mul(&zi3, &zi2, zi);
68  secp256k1_fe_mul(&r->x, &a->x, &zi2);
69  secp256k1_fe_mul(&r->y, &a->y, &zi3);
70  r->infinity = a->infinity;
71 }
72 
73 static void secp256k1_ge_set_xy(secp256k1_ge *r, const secp256k1_fe *x, const secp256k1_fe *y) {
74  r->infinity = 0;
75  r->x = *x;
76  r->y = *y;
77 }
78 
79 static int secp256k1_ge_is_infinity(const secp256k1_ge *a) {
80  return a->infinity;
81 }
82 
83 static void secp256k1_ge_neg(secp256k1_ge *r, const secp256k1_ge *a) {
84  *r = *a;
86  secp256k1_fe_negate(&r->y, &r->y, 1);
87 }
88 
90  secp256k1_fe z2, z3;
91  r->infinity = a->infinity;
92  secp256k1_fe_inv(&a->z, &a->z);
93  secp256k1_fe_sqr(&z2, &a->z);
94  secp256k1_fe_mul(&z3, &a->z, &z2);
95  secp256k1_fe_mul(&a->x, &a->x, &z2);
96  secp256k1_fe_mul(&a->y, &a->y, &z3);
97  secp256k1_fe_set_int(&a->z, 1);
98  r->x = a->x;
99  r->y = a->y;
100 }
101 
103  secp256k1_fe z2, z3;
104  r->infinity = a->infinity;
105  if (a->infinity) {
106  return;
107  }
108  secp256k1_fe_inv_var(&a->z, &a->z);
109  secp256k1_fe_sqr(&z2, &a->z);
110  secp256k1_fe_mul(&z3, &a->z, &z2);
111  secp256k1_fe_mul(&a->x, &a->x, &z2);
112  secp256k1_fe_mul(&a->y, &a->y, &z3);
113  secp256k1_fe_set_int(&a->z, 1);
114  r->x = a->x;
115  r->y = a->y;
116 }
117 
118 static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len) {
119  secp256k1_fe u;
120  size_t i;
121  size_t last_i = SIZE_MAX;
122 
123  for (i = 0; i < len; i++) {
124  if (!a[i].infinity) {
125  /* Use destination's x coordinates as scratch space */
126  if (last_i == SIZE_MAX) {
127  r[i].x = a[i].z;
128  } else {
129  secp256k1_fe_mul(&r[i].x, &r[last_i].x, &a[i].z);
130  }
131  last_i = i;
132  }
133  }
134  if (last_i == SIZE_MAX) {
135  return;
136  }
137  secp256k1_fe_inv_var(&u, &r[last_i].x);
138 
139  i = last_i;
140  while (i > 0) {
141  i--;
142  if (!a[i].infinity) {
143  secp256k1_fe_mul(&r[last_i].x, &r[i].x, &u);
144  secp256k1_fe_mul(&u, &u, &a[last_i].z);
145  last_i = i;
146  }
147  }
148  VERIFY_CHECK(!a[last_i].infinity);
149  r[last_i].x = u;
150 
151  for (i = 0; i < len; i++) {
152  r[i].infinity = a[i].infinity;
153  if (!a[i].infinity) {
154  secp256k1_ge_set_gej_zinv(&r[i], &a[i], &r[i].x);
155  }
156  }
157 }
158 
159 static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr) {
160  size_t i = len - 1;
161  secp256k1_fe zs;
162 
163  if (len > 0) {
164  /* The z of the final point gives us the "global Z" for the table. */
165  r[i].x = a[i].x;
166  r[i].y = a[i].y;
167  /* Ensure all y values are in weak normal form for fast negation of points */
169  *globalz = a[i].z;
170  r[i].infinity = 0;
171  zs = zr[i];
172 
173  /* Work our way backwards, using the z-ratios to scale the x/y values. */
174  while (i > 0) {
175  if (i != len - 1) {
176  secp256k1_fe_mul(&zs, &zs, &zr[i]);
177  }
178  i--;
179  secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zs);
180  }
181  }
182 }
183 
185  r->infinity = 1;
186  secp256k1_fe_clear(&r->x);
187  secp256k1_fe_clear(&r->y);
188  secp256k1_fe_clear(&r->z);
189 }
190 
192  r->infinity = 1;
193  secp256k1_fe_clear(&r->x);
194  secp256k1_fe_clear(&r->y);
195 }
196 
198  r->infinity = 0;
199  secp256k1_fe_clear(&r->x);
200  secp256k1_fe_clear(&r->y);
201  secp256k1_fe_clear(&r->z);
202 }
203 
205  r->infinity = 0;
206  secp256k1_fe_clear(&r->x);
207  secp256k1_fe_clear(&r->y);
208 }
209 
211  secp256k1_fe x2, x3;
212  r->x = *x;
213  secp256k1_fe_sqr(&x2, x);
214  secp256k1_fe_mul(&x3, x, &x2);
215  r->infinity = 0;
216  secp256k1_fe_add(&x3, &secp256k1_fe_const_b);
217  return secp256k1_fe_sqrt(&r->y, &x3);
218 }
219 
220 static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd) {
221  if (!secp256k1_ge_set_xquad(r, x)) {
222  return 0;
223  }
225  if (secp256k1_fe_is_odd(&r->y) != odd) {
226  secp256k1_fe_negate(&r->y, &r->y, 1);
227  }
228  return 1;
229 
230 }
231 
233  r->infinity = a->infinity;
234  r->x = a->x;
235  r->y = a->y;
236  secp256k1_fe_set_int(&r->z, 1);
237 }
238 
239 static int secp256k1_gej_eq_x_var(const secp256k1_fe *x, const secp256k1_gej *a) {
240  secp256k1_fe r, r2;
241  VERIFY_CHECK(!a->infinity);
242  secp256k1_fe_sqr(&r, &a->z); secp256k1_fe_mul(&r, &r, x);
243  r2 = a->x; secp256k1_fe_normalize_weak(&r2);
244  return secp256k1_fe_equal_var(&r, &r2);
245 }
246 
247 static void secp256k1_gej_neg(secp256k1_gej *r, const secp256k1_gej *a) {
248  r->infinity = a->infinity;
249  r->x = a->x;
250  r->y = a->y;
251  r->z = a->z;
253  secp256k1_fe_negate(&r->y, &r->y, 1);
254 }
255 
257  return a->infinity;
258 }
259 
261  secp256k1_fe y2, x3;
262  if (a->infinity) {
263  return 0;
264  }
265  /* y^2 = x^3 + 7 */
266  secp256k1_fe_sqr(&y2, &a->y);
267  secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
268  secp256k1_fe_add(&x3, &secp256k1_fe_const_b);
270  return secp256k1_fe_equal_var(&y2, &x3);
271 }
272 
274  /* Operations: 3 mul, 4 sqr, 0 normalize, 12 mul_int/add/negate.
275  *
276  * Note that there is an implementation described at
277  * https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
278  * which trades a multiply for a square, but in practice this is actually slower,
279  * mainly because it requires more normalizations.
280  */
281  secp256k1_fe t1,t2,t3,t4;
282 
283  r->infinity = a->infinity;
284 
285  secp256k1_fe_mul(&r->z, &a->z, &a->y);
286  secp256k1_fe_mul_int(&r->z, 2); /* Z' = 2*Y*Z (2) */
287  secp256k1_fe_sqr(&t1, &a->x);
288  secp256k1_fe_mul_int(&t1, 3); /* T1 = 3*X^2 (3) */
289  secp256k1_fe_sqr(&t2, &t1); /* T2 = 9*X^4 (1) */
290  secp256k1_fe_sqr(&t3, &a->y);
291  secp256k1_fe_mul_int(&t3, 2); /* T3 = 2*Y^2 (2) */
292  secp256k1_fe_sqr(&t4, &t3);
293  secp256k1_fe_mul_int(&t4, 2); /* T4 = 8*Y^4 (2) */
294  secp256k1_fe_mul(&t3, &t3, &a->x); /* T3 = 2*X*Y^2 (1) */
295  r->x = t3;
296  secp256k1_fe_mul_int(&r->x, 4); /* X' = 8*X*Y^2 (4) */
297  secp256k1_fe_negate(&r->x, &r->x, 4); /* X' = -8*X*Y^2 (5) */
298  secp256k1_fe_add(&r->x, &t2); /* X' = 9*X^4 - 8*X*Y^2 (6) */
299  secp256k1_fe_negate(&t2, &t2, 1); /* T2 = -9*X^4 (2) */
300  secp256k1_fe_mul_int(&t3, 6); /* T3 = 12*X*Y^2 (6) */
301  secp256k1_fe_add(&t3, &t2); /* T3 = 12*X*Y^2 - 9*X^4 (8) */
302  secp256k1_fe_mul(&r->y, &t1, &t3); /* Y' = 36*X^3*Y^2 - 27*X^6 (1) */
303  secp256k1_fe_negate(&t2, &t4, 2); /* T2 = -8*Y^4 (3) */
304  secp256k1_fe_add(&r->y, &t2); /* Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4) */
305 }
306 
318  if (a->infinity) {
319  r->infinity = 1;
320  if (rzr != NULL) {
321  secp256k1_fe_set_int(rzr, 1);
322  }
323  return;
324  }
325 
326  if (rzr != NULL) {
327  *rzr = a->y;
329  secp256k1_fe_mul_int(rzr, 2);
330  }
331 
332  secp256k1_gej_double(r, a);
333 }
334 
336  /* Operations: 12 mul, 4 sqr, 2 normalize, 12 mul_int/add/negate */
337  secp256k1_fe z22, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
338 
339  if (a->infinity) {
340  VERIFY_CHECK(rzr == NULL);
341  *r = *b;
342  return;
343  }
344 
345  if (b->infinity) {
346  if (rzr != NULL) {
347  secp256k1_fe_set_int(rzr, 1);
348  }
349  *r = *a;
350  return;
351  }
352 
353  r->infinity = 0;
354  secp256k1_fe_sqr(&z22, &b->z);
355  secp256k1_fe_sqr(&z12, &a->z);
356  secp256k1_fe_mul(&u1, &a->x, &z22);
357  secp256k1_fe_mul(&u2, &b->x, &z12);
358  secp256k1_fe_mul(&s1, &a->y, &z22); secp256k1_fe_mul(&s1, &s1, &b->z);
359  secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
360  secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
361  secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
364  secp256k1_gej_double_var(r, a, rzr);
365  } else {
366  if (rzr != NULL) {
367  secp256k1_fe_set_int(rzr, 0);
368  }
370  }
371  return;
372  }
373  secp256k1_fe_sqr(&i2, &i);
374  secp256k1_fe_sqr(&h2, &h);
375  secp256k1_fe_mul(&h3, &h, &h2);
376  secp256k1_fe_mul(&h, &h, &b->z);
377  if (rzr != NULL) {
378  *rzr = h;
379  }
380  secp256k1_fe_mul(&r->z, &a->z, &h);
381  secp256k1_fe_mul(&t, &u1, &h2);
382  r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
383  secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
384  secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
385  secp256k1_fe_add(&r->y, &h3);
386 }
387 
389  /* 8 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
390  secp256k1_fe z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
391  if (a->infinity) {
392  VERIFY_CHECK(rzr == NULL);
393  secp256k1_gej_set_ge(r, b);
394  return;
395  }
396  if (b->infinity) {
397  if (rzr != NULL) {
398  secp256k1_fe_set_int(rzr, 1);
399  }
400  *r = *a;
401  return;
402  }
403  r->infinity = 0;
404 
405  secp256k1_fe_sqr(&z12, &a->z);
406  u1 = a->x; secp256k1_fe_normalize_weak(&u1);
407  secp256k1_fe_mul(&u2, &b->x, &z12);
408  s1 = a->y; secp256k1_fe_normalize_weak(&s1);
409  secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
410  secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
411  secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
414  secp256k1_gej_double_var(r, a, rzr);
415  } else {
416  if (rzr != NULL) {
417  secp256k1_fe_set_int(rzr, 0);
418  }
420  }
421  return;
422  }
423  secp256k1_fe_sqr(&i2, &i);
424  secp256k1_fe_sqr(&h2, &h);
425  secp256k1_fe_mul(&h3, &h, &h2);
426  if (rzr != NULL) {
427  *rzr = h;
428  }
429  secp256k1_fe_mul(&r->z, &a->z, &h);
430  secp256k1_fe_mul(&t, &u1, &h2);
431  r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
432  secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
433  secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
434  secp256k1_fe_add(&r->y, &h3);
435 }
436 
437 static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, const secp256k1_fe *bzinv) {
438  /* 9 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
439  secp256k1_fe az, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
440 
441  if (b->infinity) {
442  *r = *a;
443  return;
444  }
445  if (a->infinity) {
446  secp256k1_fe bzinv2, bzinv3;
447  r->infinity = b->infinity;
448  secp256k1_fe_sqr(&bzinv2, bzinv);
449  secp256k1_fe_mul(&bzinv3, &bzinv2, bzinv);
450  secp256k1_fe_mul(&r->x, &b->x, &bzinv2);
451  secp256k1_fe_mul(&r->y, &b->y, &bzinv3);
452  secp256k1_fe_set_int(&r->z, 1);
453  return;
454  }
455  r->infinity = 0;
456 
465  secp256k1_fe_mul(&az, &a->z, bzinv);
466 
467  secp256k1_fe_sqr(&z12, &az);
468  u1 = a->x; secp256k1_fe_normalize_weak(&u1);
469  secp256k1_fe_mul(&u2, &b->x, &z12);
470  s1 = a->y; secp256k1_fe_normalize_weak(&s1);
471  secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &az);
472  secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
473  secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
476  secp256k1_gej_double_var(r, a, NULL);
477  } else {
479  }
480  return;
481  }
482  secp256k1_fe_sqr(&i2, &i);
483  secp256k1_fe_sqr(&h2, &h);
484  secp256k1_fe_mul(&h3, &h, &h2);
485  r->z = a->z; secp256k1_fe_mul(&r->z, &r->z, &h);
486  secp256k1_fe_mul(&t, &u1, &h2);
487  r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
488  secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
489  secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
490  secp256k1_fe_add(&r->y, &h3);
491 }
492 
493 
494 static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b) {
495  /* Operations: 7 mul, 5 sqr, 4 normalize, 21 mul_int/add/negate/cmov */
496  static const secp256k1_fe fe_1 = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
497  secp256k1_fe zz, u1, u2, s1, s2, t, tt, m, n, q, rr;
498  secp256k1_fe m_alt, rr_alt;
499  int infinity, degenerate;
500  VERIFY_CHECK(!b->infinity);
501  VERIFY_CHECK(a->infinity == 0 || a->infinity == 1);
502 
553  secp256k1_fe_sqr(&zz, &a->z); /* z = Z1^2 */
554  u1 = a->x; secp256k1_fe_normalize_weak(&u1); /* u1 = U1 = X1*Z2^2 (1) */
555  secp256k1_fe_mul(&u2, &b->x, &zz); /* u2 = U2 = X2*Z1^2 (1) */
556  s1 = a->y; secp256k1_fe_normalize_weak(&s1); /* s1 = S1 = Y1*Z2^3 (1) */
557  secp256k1_fe_mul(&s2, &b->y, &zz); /* s2 = Y2*Z1^2 (1) */
558  secp256k1_fe_mul(&s2, &s2, &a->z); /* s2 = S2 = Y2*Z1^3 (1) */
559  t = u1; secp256k1_fe_add(&t, &u2); /* t = T = U1+U2 (2) */
560  m = s1; secp256k1_fe_add(&m, &s2); /* m = M = S1+S2 (2) */
561  secp256k1_fe_sqr(&rr, &t); /* rr = T^2 (1) */
562  secp256k1_fe_negate(&m_alt, &u2, 1); /* Malt = -X2*Z1^2 */
563  secp256k1_fe_mul(&tt, &u1, &m_alt); /* tt = -U1*U2 (2) */
564  secp256k1_fe_add(&rr, &tt); /* rr = R = T^2-U1*U2 (3) */
567  degenerate = secp256k1_fe_normalizes_to_zero(&m) &
569  /* This only occurs when y1 == -y2 and x1^3 == x2^3, but x1 != x2.
570  * This means either x1 == beta*x2 or beta*x1 == x2, where beta is
571  * a nontrivial cube root of one. In either case, an alternate
572  * non-indeterminate expression for lambda is (y1 - y2)/(x1 - x2),
573  * so we set R/M equal to this. */
574  rr_alt = s1;
575  secp256k1_fe_mul_int(&rr_alt, 2); /* rr = Y1*Z2^3 - Y2*Z1^3 (2) */
576  secp256k1_fe_add(&m_alt, &u1); /* Malt = X1*Z2^2 - X2*Z1^2 */
577 
578  secp256k1_fe_cmov(&rr_alt, &rr, !degenerate);
579  secp256k1_fe_cmov(&m_alt, &m, !degenerate);
580  /* Now Ralt / Malt = lambda and is guaranteed not to be 0/0.
581  * From here on out Ralt and Malt represent the numerator
582  * and denominator of lambda; R and M represent the explicit
583  * expressions x1^2 + x2^2 + x1x2 and y1 + y2. */
584  secp256k1_fe_sqr(&n, &m_alt); /* n = Malt^2 (1) */
585  secp256k1_fe_mul(&q, &n, &t); /* q = Q = T*Malt^2 (1) */
586  /* These two lines use the observation that either M == Malt or M == 0,
587  * so M^3 * Malt is either Malt^4 (which is computed by squaring), or
588  * zero (which is "computed" by cmov). So the cost is one squaring
589  * versus two multiplications. */
590  secp256k1_fe_sqr(&n, &n);
591  secp256k1_fe_cmov(&n, &m, degenerate); /* n = M^3 * Malt (2) */
592  secp256k1_fe_sqr(&t, &rr_alt); /* t = Ralt^2 (1) */
593  secp256k1_fe_mul(&r->z, &a->z, &m_alt); /* r->z = Malt*Z (1) */
594  infinity = secp256k1_fe_normalizes_to_zero(&r->z) * (1 - a->infinity);
595  secp256k1_fe_mul_int(&r->z, 2); /* r->z = Z3 = 2*Malt*Z (2) */
596  secp256k1_fe_negate(&q, &q, 1); /* q = -Q (2) */
597  secp256k1_fe_add(&t, &q); /* t = Ralt^2-Q (3) */
599  r->x = t; /* r->x = Ralt^2-Q (1) */
600  secp256k1_fe_mul_int(&t, 2); /* t = 2*x3 (2) */
601  secp256k1_fe_add(&t, &q); /* t = 2*x3 - Q: (4) */
602  secp256k1_fe_mul(&t, &t, &rr_alt); /* t = Ralt*(2*x3 - Q) (1) */
603  secp256k1_fe_add(&t, &n); /* t = Ralt*(2*x3 - Q) + M^3*Malt (3) */
604  secp256k1_fe_negate(&r->y, &t, 3); /* r->y = Ralt*(Q - 2x3) - M^3*Malt (4) */
606  secp256k1_fe_mul_int(&r->x, 4); /* r->x = X3 = 4*(Ralt^2-Q) */
607  secp256k1_fe_mul_int(&r->y, 4); /* r->y = Y3 = 4*Ralt*(Q - 2x3) - 4*M^3*Malt (4) */
608 
610  secp256k1_fe_cmov(&r->x, &b->x, a->infinity);
611  secp256k1_fe_cmov(&r->y, &b->y, a->infinity);
612  secp256k1_fe_cmov(&r->z, &fe_1, a->infinity);
613  r->infinity = infinity;
614 }
615 
617  /* Operations: 4 mul, 1 sqr */
618  secp256k1_fe zz;
620  secp256k1_fe_sqr(&zz, s);
621  secp256k1_fe_mul(&r->x, &r->x, &zz); /* r->x *= s^2 */
622  secp256k1_fe_mul(&r->y, &r->y, &zz);
623  secp256k1_fe_mul(&r->y, &r->y, s); /* r->y *= s^3 */
624  secp256k1_fe_mul(&r->z, &r->z, s); /* r->z *= s */
625 }
626 
628  secp256k1_fe x, y;
629  VERIFY_CHECK(!a->infinity);
630  x = a->x;
632  y = a->y;
634  secp256k1_fe_to_storage(&r->x, &x);
635  secp256k1_fe_to_storage(&r->y, &y);
636 }
637 
639  secp256k1_fe_from_storage(&r->x, &a->x);
640  secp256k1_fe_from_storage(&r->y, &a->y);
641  r->infinity = 0;
642 }
643 
645  secp256k1_fe_storage_cmov(&r->x, &a->x, flag);
646  secp256k1_fe_storage_cmov(&r->y, &a->y, flag);
647 }
648 
650  static const secp256k1_fe beta = SECP256K1_FE_CONST(
651  0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul,
652  0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul
653  );
654  *r = *a;
655  secp256k1_fe_mul(&r->x, &r->x, &beta);
656 }
657 
659  secp256k1_fe yz;
660 
661  if (a->infinity) {
662  return 0;
663  }
664 
665  /* We rely on the fact that the Jacobi symbol of 1 / a->z^3 is the same as
666  * that of a->z. Thus a->y / a->z^3 is a quadratic residue iff a->y * a->z
667  is */
668  secp256k1_fe_mul(&yz, &a->y, &a->z);
669  return secp256k1_fe_is_quad_var(&yz);
670 }
671 
673 #ifdef EXHAUSTIVE_TEST_ORDER
674  secp256k1_gej out;
675  int i;
676 
677  /* A very simple EC multiplication ladder that avoids a dependecy on ecmult. */
679  for (i = 0; i < 32; ++i) {
680  secp256k1_gej_double_var(&out, &out, NULL);
681  if ((((uint32_t)EXHAUSTIVE_TEST_ORDER) >> (31 - i)) & 1) {
682  secp256k1_gej_add_ge_var(&out, &out, ge, NULL);
683  }
684  }
685  return secp256k1_gej_is_infinity(&out);
686 #else
687  (void)ge;
688  /* The real secp256k1 group has cofactor 1, so the subgroup is the entire curve. */
689  return 1;
690 #endif
691 }
692 
693 #endif /* SECP256K1_GROUP_IMPL_H */
#define VERIFY_CHECK(cond)
Definition: util.h:68
static int secp256k1_fe_is_zero(const secp256k1_fe *a)
Verify whether a field element is zero.
static void secp256k1_fe_mul(secp256k1_fe *r, const secp256k1_fe *a, const secp256k1_fe *SECP256K1_RESTRICT b)
Sets a field element to be the product of two others.
static void secp256k1_fe_normalize_var(secp256k1_fe *r)
Normalize a field element, without constant-time guarantee.
secp256k1_fe x
Definition: group.h:25
static void secp256k1_gej_rescale(secp256k1_gej *r, const secp256k1_fe *s)
Definition: group_impl.h:616
static void secp256k1_ge_set_gej_var(secp256k1_ge *r, secp256k1_gej *a)
Definition: group_impl.h:102
static void secp256k1_ge_neg(secp256k1_ge *r, const secp256k1_ge *a)
Definition: group_impl.h:83
static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr)
Definition: group_impl.h:307
static void secp256k1_fe_negate(secp256k1_fe *r, const secp256k1_fe *a, int m)
Set a field element equal to the additive inverse of another.
static void secp256k1_fe_from_storage(secp256k1_fe *r, const secp256k1_fe_storage *a)
Convert a field element back from the storage type.
static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, secp256k1_fe *rzr)
Definition: group_impl.h:388
static void secp256k1_gej_clear(secp256k1_gej *r)
Definition: group_impl.h:197
static void secp256k1_fe_storage_cmov(secp256k1_fe_storage *r, const secp256k1_fe_storage *a, int flag)
If flag is true, set *r equal to *a; otherwise leave it.
static void secp256k1_fe_cmov(secp256k1_fe *r, const secp256k1_fe *a, int flag)
If flag is true, set *r equal to *a; otherwise leave it.
secp256k1_fe_storage y
Definition: group.h:36
static void secp256k1_ge_set_gej(secp256k1_ge *r, secp256k1_gej *a)
Definition: group_impl.h:89
A group element of the secp256k1 curve, in jacobian coordinates.
Definition: group.h:24
static void secp256k1_fe_set_int(secp256k1_fe *r, int a)
Set a field element equal to a small integer.
static void secp256k1_fe_to_storage(secp256k1_fe_storage *r, const secp256k1_fe *a)
Convert a field element to the storage type.
static int secp256k1_gej_has_quad_y_var(const secp256k1_gej *a)
Definition: group_impl.h:658
#define SECP256K1_FE_CONST(d7, d6, d5, d4, d3, d2, d1, d0)
Definition: field_10x26.h:40
static void secp256k1_fe_clear(secp256k1_fe *a)
Sets a field element equal to zero, initializing all fields.
static void secp256k1_fe_add(secp256k1_fe *r, const secp256k1_fe *a)
Adds a field element to another.
static void secp256k1_fe_mul_int(secp256k1_fe *r, int a)
Multiplies the passed field element with a small integer constant.
static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b)
Definition: group_impl.h:494
static int secp256k1_ge_is_infinity(const secp256k1_ge *a)
Definition: group_impl.h:79
static int secp256k1_fe_is_odd(const secp256k1_fe *a)
Check the "oddness" of a field element.
static void secp256k1_gej_set_ge(secp256k1_gej *r, const secp256k1_ge *a)
Definition: group_impl.h:232
static int secp256k1_gej_eq_x_var(const secp256k1_fe *x, const secp256k1_gej *a)
Definition: group_impl.h:239
static int secp256k1_ge_is_in_correct_subgroup(const secp256k1_ge *ge)
Definition: group_impl.h:672
static const secp256k1_ge secp256k1_ge_const_g
Generator for secp256k1, value &#39;g&#39; defined in "Standards for Efficient Cryptography" (SEC2) 2...
Definition: group_impl.h:53
static void secp256k1_ge_from_storage(secp256k1_ge *r, const secp256k1_ge_storage *a)
Definition: group_impl.h:638
static void secp256k1_ge_set_xy(secp256k1_ge *r, const secp256k1_fe *x, const secp256k1_fe *y)
Definition: group_impl.h:73
static void secp256k1_gej_set_infinity(secp256k1_gej *r)
Definition: group_impl.h:184
static void secp256k1_gej_neg(secp256k1_gej *r, const secp256k1_gej *a)
Definition: group_impl.h:247
#define SECP256K1_INLINE
Definition: secp256k1.h:124
int infinity
Definition: group.h:28
static int secp256k1_ge_set_xquad(secp256k1_ge *r, const secp256k1_fe *x)
Definition: group_impl.h:210
secp256k1_fe_storage x
Definition: group.h:35
static int secp256k1_fe_is_quad_var(const secp256k1_fe *a)
Checks whether a field element is a quadratic residue.
static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a)
Definition: group_impl.h:649
static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len)
Definition: group_impl.h:118
static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd)
Definition: group_impl.h:220
A group element of the secp256k1 curve, in affine coordinates.
Definition: group.h:14
secp256k1_fe x
Definition: group.h:15
static void secp256k1_fe_normalize_weak(secp256k1_fe *r)
Weakly normalize a field element: reduce its magnitude to 1, but don&#39;t fully normalize.
static int secp256k1_fe_normalizes_to_zero(secp256k1_fe *r)
Verify whether a field element represents zero i.e.
int infinity
Definition: group.h:17
static void secp256k1_ge_set_infinity(secp256k1_ge *r)
Definition: group_impl.h:191
static void secp256k1_fe_sqr(secp256k1_fe *r, const secp256k1_fe *a)
Sets a field element to be the square of another.
#define SECP256K1_GE_CONST(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p)
Definition: group.h:20
static const secp256k1_fe secp256k1_fe_const_b
Definition: group_impl.h:60
static SECP256K1_INLINE void secp256k1_ge_storage_cmov(secp256k1_ge_storage *r, const secp256k1_ge_storage *a, int flag)
Definition: group_impl.h:644
static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, const secp256k1_fe *bzinv)
Definition: group_impl.h:437
static int secp256k1_fe_equal_var(const secp256k1_fe *a, const secp256k1_fe *b)
Same as secp256k1_fe_equal, but may be variable time.
static void secp256k1_ge_to_storage(secp256k1_ge_storage *r, const secp256k1_ge *a)
Definition: group_impl.h:627
#define EXHAUSTIVE_TEST_ORDER
static int secp256k1_gej_is_infinity(const secp256k1_gej *a)
Definition: group_impl.h:256
static void secp256k1_ge_set_gej_zinv(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zi)
Definition: group_impl.h:63
secp256k1_fe z
Definition: group.h:27
static SECP256K1_INLINE void secp256k1_gej_double(secp256k1_gej *r, const secp256k1_gej *a)
Definition: group_impl.h:273
static void secp256k1_fe_normalize(secp256k1_fe *r)
Field element module.
static int secp256k1_ge_is_valid_var(const secp256k1_ge *a)
Definition: group_impl.h:260
static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr)
Definition: group_impl.h:159
secp256k1_fe y
Definition: group.h:26
static int secp256k1_fe_sqrt(secp256k1_fe *r, const secp256k1_fe *a)
If a has a square root, it is computed in r and 1 is returned.
secp256k1_fe y
Definition: group.h:16
static void secp256k1_fe_inv_var(secp256k1_fe *r, const secp256k1_fe *a)
Potentially faster version of secp256k1_fe_inv, without constant-time guarantee.
static void secp256k1_fe_inv(secp256k1_fe *r, const secp256k1_fe *a)
Sets a field element to be the (modular) inverse of another.
static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_gej *b, secp256k1_fe *rzr)
Definition: group_impl.h:335
static void secp256k1_ge_clear(secp256k1_ge *r)
Definition: group_impl.h:204
static int secp256k1_fe_normalizes_to_zero_var(secp256k1_fe *r)
Verify whether a field element represents zero i.e.